Boundedness of Commutators on Hardy Spaces over Metric Measure Spaces of Non-homogeneous Type

Abstract: Let $(\mathcal{X},d,\mu)$ be a metric measure space satisfying the so-called upper doubling condition and the geometrically doubling condition. Let $T$ be a Calder\'{o}n-Zygmund operator with kernel satisfying only the size condition and some H\"ormander-type condition, and $b\in\rm{\widetilde{RBMO}(\mu)}$ (the regularized BMO space with the discrete coefficient). In this paper, the authors establish the boundedness of the commutator $T_b:=bT-Tb$ generated by $T$ and $b$ from the atomic Hardy space $\widetilde{H}^1(\mu)$ with the discrete coefficient into the weak Lebesgue space $L^{1,\,\infty}(\mu)$. The boundedness of the commutator generated by the generalized fractional integral $T_\alpha\,(\alpha\in(0,1))$ and the $\rm{\widetilde{RBMO}(\mu)}$ function from $\widetilde{H}^1(\mu)$ into $L^{1/{(1-\alpha)},\,\fz}(\mu)$ is also presented. Moreover, by an interpolation theorem for sublinear operators, the authors show that the commutator $T_b$ is bounded on $L^p(\mu)$ for all $p\in(1,\infty)$.